Heretofore, a flow rate control apparatus has been known, in which a pressure fluid supplied from a pressure fluid supply source is controlled at a predetermined flow rate, and is supplied to a fluid pressure device or the like connected to a secondary side. With such a flow rate control apparatus, for example, as disclosed in European Patent No. 2047108, a soft start technique is adopted in which, initially, a pressure fluid is made to flow through a throttle valve in which the flow rate of the pressure fluid is restricted, and after elapse of a predetermined time, the valve body is switched to an open state and the flow rate of the pressure fluid is increased. By means of this structure, abrupt supply of a pressure fluid having a large flow rate with respect to a fluid pressure device connected to a secondary side of the flow rate control apparatus, thereby causing damage to the fluid pressure device, is prevented.